What Are Commutative, Associative and Distributive Properties?

The commutative, associative and distributive properties describe how basic mathematical operations work. The properties are helpful in finding efficient ways to solve equations and in simplifying algebraic expressions.

Addition and multiplication have the commutative property, meaning that numbers can be added or multiplied together in any order without affecting the result. In other words, adding 7 + 3 is the same as adding 3 + 7. Multiplying 2 by 4.5 has the same outcome as multiplying 4.5 by 2. The commutative property for addition is expressed as a + b = b + a. The commutative property for multiplication is expressed as a * b = b * a.

Addition and multiplication also have the associative property, meaning that numbers can be added or multiplied in any grouping (or association) without affecting the result. For example, (3 + 2) + 7 has the same result as 3 + (2 + 7), while (4 * 2) * 5 has the same result as 4 * (2 * 5). The associative property for addition is expressed as (a + b) + c = a + (b + c). The associative property for multiplication is expressed as (a * b) * c = a * (b * c).

Only multiplication has the distributive property, which applies to expressions that multiply a number by a sum or difference. Multiplication distributes over addition because a(b + c) = ab + ac. For example, to multiply 2 by the sum of 9 + 4, the numbers 9 and 4 can be added first to find the sum of 13, and then 13 can be multiplied by 2 to return 26. A different way to achieve the same result is to distribute the 2 by first multiplying 2 by 9 (18) and 2 by 4 (8). The two results can be added (18 + 8) to return 26 as before. In the same way, multiplication distributes over subtraction because a(b – c) = ab – ac.